Zeroes of Sine and Cosine

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Theorem

$(1): \quad \forall n \in \Z: x = \paren {n + \dfrac 1 2} \pi \implies \cos x = 0$
$(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$


Proof

From Sine and Cosine are Periodic on Reals: Corollary:


$\cos x$ is:

strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$

and:

strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$


$\sin x$ is:

strictly positive on the interval $\openint 0 \pi$

and:

strictly negative on the interval $\openint \pi {2 \pi}$


The result follows directly from Sine and Cosine are Periodic on Reals.

$\blacksquare$